To find the third term in the expansion of \((a+b)^5\) using Pascal's triangle, you first need to identify the coefficients from the triangle for the corresponding row.
The coefficients for \((a+b)^n\) can be found in the \(n\)th row of Pascal's triangle, where \(n\) is the exponent. For \(n = 5\), the coefficients are:
\[ 1, 5, 10, 10, 5, 1 \]
The general term in the expansion of \((a+b)^n\) is given by:
\[ \text{Term } k = \binom{n}{k} a^{n-k} b^k \]
Where \(\binom{n}{k}\) is the binomial coefficient, and \(k\) starts at 0.
To find the third term, we set \(k = 2\) (since we start counting terms from \(k = 0\)):
\[ \text{Term } 2 = \binom{5}{2} a^{5-2} b^{2} \]
Calculating this:
- The binomial coefficient \(\binom{5}{2} = 10\).
- The term then becomes: \[ 10 a^{3} b^{2} \]
Thus, the third term in the expansion of \((a + b)^5\) is:
\[ 10 a^{3} b^{2} \]
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